Polarimetry of Black Hole Spin and Axisymmetric Emission Geometry

The Event Horizon Telescope array has measured the average polarimetric spiral morphology of black hole accretion disks using the so-called ``${\beta }_2$’’ coefficient. This measurement is dominated by the weakly lensed, direct ($n=0$, counting photon half-orbits) image of the disk, providing insight into magnetic field structure around the M87* black hole. The future space-based Black Hole Explorer mission is expected to extend these capabilities, allowing for ${\beta }_2$ measurement of the strongly lensed $n=1$ image. In this paper, we explore the constraints ${\beta }_2$ measurements would impose on spacetime and plasma geometries using a semi-analytic conical emission model. We show that magnetic field orientation with respect to the emission cone qualitatively changes the dependence of ${\beta }_2$ modes on model parameters. For cone-aligned magnetic fields, we find that the leading-order theoretical prediction from equatorial emitters of ${\beta }_{2,n=0}\approx {\beta }_{2,n=1}^{\ast }$ is the dominant trend even in off-equatorial models. Specifically, we show $\arg ({\beta }_{2,n=0}{\beta }_{2,n=1}^{\ast })\approx 4\chi$, where $\chi$ is the angle determining in-cone magnetic field orientation. We also find that $\arg ({\beta }_{2,n=0}{\beta }_{2,n=1})$ minimizes the effect of magnetic field geometry while preserving a dependence on spin that may be used to measure spin direction. We find that observations of the photon ring’s ${\beta }_2$ complement the constraint from direct image polarization, and can greatly constrain magnetic field structure even with an unknown external rotation measure. We also find that black hole spin and emission cone opening angle would not be tightly constrained by ${\beta }_2$ alone, unless strong priors are placed on the magnetic field geometry.